First off, this guide assumes that you've already picked out your ships, the ore you want, and that you know the general mechanics of mining and hauling. If you're a complete newbie to mining, I have to recommend this huge PDF guide to everything mining in EVE Online by Halada. It's a big document, but it'll get you going from when you first start skilling out, to how to survive as a miner in lowsec space. So let's assume you've read that, and you've gotten a ship, a hauler (a ship with a larger cargohold to haul bigger quantities of ore), and know the ore that you're going after and where to find it.
Grab your smaller ship fitted with mining lasers, head out to the asteroid field, and find the asteroid you want to mine. Now you're ready to start jetcanning.
Jetcanning is basically the process of storing up a bunch of ore over time in what's called a "jetcan" (short for jettisoned canister), and then hauling it back to your homebase (or wherever you choose to refine or sell it). If you've done any mining in EVE at all, you've probably already figured out that the most expensive thing you can do in terms of time (which as we all know, is money) is traveling, so the idea behind jetcanning is that the less you can travel, the better.
What you'll do is go out to the asteroid you want to mine, flip on whatever lasers you have equipped, and start pulling in ore. As you might know, the rate of ore you pull in depends on many, many things-- what lasers you're using (usually Miner I or Miner II, although at the higher levels, there are more options), what ship you're using (ships like the Osprey have a mining bonus), what skills you've trained (obviously Mining is important), and so on. But no matter what rate you've reached, ore will start to fill up your ship's hold as long as you're within range of an asteroid and have a laser targeted on it and breaking it apart.
When it does, you'll want to right click on the ore, and choose "Jettison." This won't throw the ore away forever-- instead, it'll kick the ore out into a canister called a "Cargo Container," that you'll be able to see in space and on your Overview. If you right click on the can to Open Cargo, you'll be able to see the ore that you just jettisoned out of your ship. And here's the key-- while you've got that can open, you can put other things into it.
Most lowbie ships don't have real big holds-- even with Expanded Cargoholds, you can only hold a few thousand m3 of ore, and filling up just your hold and transferring it back to your homebase every time would take quite a while. But once you've opened up the jetcan, check out how much space is in there. That's right-- a whopping 27,500 m3. So with jetcan mining, the idea is to take a smaller ship (with mining bonuses) out, put ore into a jetcan until it fills up, and then come back with a bigger ship and haul the whole thing in only a few trips. As the ore fills up your ship's hold, just drag it into the open jetcan, and repeat until the jetcan is full. You might want to listen to music or watch some TV while you do this-- mining in deep space can be boring at times. But don't let your attention stray too far, because there are a few catches.
Jetcanning isn't the end all, be all of mining, either-- eventually you'll probably train up to ships so big that you don't need to jettison the ore to hold it, and then, at the highest levels, you might not even need to take the ore back to base at all (you'll be able to refine it right on your ship). But for a beginning miner, jetcanning is probably the fastest and quickest way to pull in a lot of ore. You can jetcan in groups, too-- have a few players pour into their own jetcans (or one big one, although more experienced miners will probably fill a little faster than beginners), and then have one player haul everything back to base, with everyone taking a cut of the profits.